arXiv subjects

Arnaud Vanhaecke

Publications and source records attributed to Arnaud Vanhaecke.

4 recordsLinked to original sources

arXiv2026

On Drinfeld's representability theorem

In the seventies, V. G. Drinfeld proved that a moduli problem of deformations by quasi-isogenies of certain $p$-divisible groups with extra actions is representable by an explicit semi-stable model of the $p$-adic symmetric space. This theorem, known as \emph{Drinfeld's representability theorem}, has been one of the cornerstones of geometric aspects in $p$-adic Hodge theory. The purpose of these notes is twofold. On the one hand we give a new and more transparent proof of Drinfeld's representability theorem; on the other hand, we give a detailed presentation of Drinfeld's moduli space and the formal model of the $p$-adic symmetric space.

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arXiv2025

Factorisation de la cohomologie de systèmes locaux $p$-adiques sur le demi-plan de Drinfeld

We compute the first cohomology group of the symmetric algebra of the universal étale $p$-adic local system on the tower of coverings of Drinfeld's $p$-adic half-plane. The result takes a factorized form, using the $p$-adic Langlands correspondence in families over Kisin rings. This work extends the corresponding results of Colmez, Dospinescu, and Niziol for trivial coefficients. It relies on the computation of automorphic multiplicities in the étale cohomology group of the local system, done in a previous paper, as well as on the determination of the Kisin rings for the special type as functions on an analytic open subset of the projective line.

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arXiv2024

Cohomologie de systèmes locaux $p$-adiques sur les revêtements du demi-plan de Drinfeld

Colmez, Dospinescu and Niziol have shown that the only $p$-adic representations of $\rm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ appearing in the $p$-adic étale cohomology of the coverings of Drinfeld's half-plane are the $2$-dimensional cuspidal representations (i.e. potentially semi-stable, whose associated Weil-Deligne representation is irreducible) with Hodge-Tate weights $0$ and $1$ and their multiplicities are given by the $p$-adic Langlands correspondence. We generalise this result to arbitrary weights, by considering the $p$-adic étale cohomology with coefficients in the symmetric powers of the universal local system on Drinfeld's tower. A novelty is the appearance of potentially semistable $2$-dimensional non-cristabelian representations, with expected multiplicity. The key point is that the local systems we consider turn out to be particularly simple: they are "isotrivial opers" on a curve. We develop a recipe to compute the proétale cohomology of such a local system using the Hyodo-Kato cohomology of the curve and the de Rham complex of the flat filtered bundle associated to the local system.

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arXiv2019

Le cristal de Dieudonné des schémas en $\mathbb{F}$-vectoriels

In this paper we describe the Dieudonné crystal of a finite locally free group scheme with a vector action of a finite field $\mathbb{F}$. These $\mathbb{F}$-vector schemes appear when we consider torsion points of $p$-divisible modules. A particular class of $\mathbb{F}$-vector schemes has been classified by Raynaud in [Ray74], which allows us to determine the structure of torsion points of $p$-divisible modules, under certain conditions on the Lie algebra.

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